Folded Preconditioner: A New Class of Preconditioners for Krylov Subspace Methods to Solve Redundancy-Reduced Linear Systems of Equations

Abstract

The A-phi formulation, which is widely used in electromagnetic analysis, leads to a redundant linear system of equations that includes a substantial number of redundant degrees of freedom (DOF). We can derive a redundancy-reduced linear system of equations by eliminating the redundant DOF, thereby decreasing the computation costs per iteration for iterative solvers, such as the incomplete Cholesky conjugate gradient (ICCG) solver. This does not, however, result in a reduction in total computation time, due to significant convergence deterioration. In this paper, we present a solution to this problem in the form of folded preconditioners. First, the theorem presented reveals that, for any preconditioned Krylov subspace method for the original redundant linear systems, we can derive the equivalent Krylov subspace method for the redundancy-reduced linear systems by using the corresponding folded preconditioner. As an uncomplicated example, the standard ICCG solver for the original redundant systems has exactly the same convergence property as the CG solver for the redundancy-reduced systems using the folded variant of the IC preconditioner (the folded IC preconditioner). Furthermore, we discuss efficient computational procedures for the folded preconditioners and the design of Krylov subspace algorithms using the preconditioners. A sample full-wave analysis demonstrates the good performance of a newly developed solver, the conjugate orthogonal conjugate gradient (COCG) method with the folded IC preconditioner. The new solver not only lowers the computation costs per iteration by reducing the number of DOF, but also completely avoids the convergence deterioration.